Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra

We continue the study of quantum matrix algebras of the GL(m|n) type. We find three alternative forms of the Cayley-Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions s _λ (M) that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood-Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 14–46, April, 2006.     

Low-frequency noise in monodisperse platinum nanostructures near the percolation threshold

The percolation threshold p ₀ ≈ 0.6 is determined for monodisperse platinum nanostructures with 1.8-nm metallic particles deposited in a monolayer onto an insulating substrate through laser electrodispersion. It is shown that, in the “metallic” state (for p > p ₀), both the magnitude of the noise and its temperature dependence are close to those of pure metallic Pt layers. The frequency dependence of the normalized noise power spectral density is described by the relationship S _I /I ² ～ 1/f ^γ with the exponent γ close to unity. For current densities j ≥ 10⁷?10⁸ A/cm², the noise power spectral density S _I increases more rapidly with a further increase in the current as compared to I ² because of the current generating excess defects. For p < p ₀, the dependence of the conductivity σ on the temperature is adequately described by the standard relationship σ ～ exp[?(T ₀/T)^1/2]. The normalized noise power spectral density S _I /I ² exceeds the corresponding value for a quasi-metallic structure by many orders of magnitude. The noise power spectral density S _I is approximately proportional to the square of the current only for very low currents and increases steeply with a further increase in the current.     

Effects of current pulses on dislocation mobility in Zn at 77 K

Selective etching at 77 K has been applied to the mobilities of pyramidal dislocations in the 11¯22<11¯23> slip system for pure Zn single crystals in the region of thermally activated motion in order to establish how the pulses affect the plasticity. The activation and statistical parameters in the dislocation motion have been related to the current density and loading on the basis of the ponderomotor forces, the thermal response and the skin effects, as well as the electron-dislocation interaction. The current effect is related to reduced probability of overcoming Peierls barriers.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 35–39, March, 1990.We are indebted to V. F. Kurilov for a discussion and critical comments, and to F. F. Lavrent'ev for providing the Zn single crystals.     

Parametric excitation of spin waves in uniaxial ferrites

Parametric excitation of spin waves in uniaxial ferrites with large anisotropy is investigated theoretically. First-order processes in a sphere with arbitrary orientation of an external static magnetic field, in which case the pumping is oblique, are studied. The threshold field and the parameters of the excited spin waves are calculated numerically for two ferrites — easy-axis and easy-plane. Zh. Tekh. Fiz. 68, 65–71 (May 1998)     

The geometric Weil representation

We construct a geometric analogue of the Weil representation over a finite field. Our construction is principally invariant, not choosing any specific realization. This eliminates most of the unpleasant formulas that appear in the traditional (non-invariant) approaches, and puts in the forefront some delicate geometric phenomena which underlie this representation.     

Mutual entrainment of hot electrons and optical phonons in polar semiconductors

Kinetic equations for nonequilibrium electrons and optical phonons are constructed and solved for the case in which the interaction between these particles is the primary mechanism for the relaxation of the electron energy and quasimomentum. The calculations reflect the circumstance that for the optical phonons the equivalent primary relaxation mechanism is the interaction with acoustic phonons (which are at equilibrium in this case). Constitutive equations are derived for polar semiconductors which reflect the mutual entrainment of electrons and optical phonons. Energy balance equations, which determine the temperatures of these particles, are also derived. These temperatures are generally different from each other and from the reservoir temperature.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 32–36, February, 1984.We wish to thank É. A. Kaner and I. B. Levinson for useful discussions.     

The theory of phonon relaxation in He(II) at low temperatures

We have investigated analytically relaxation processes in the phonon system in helium(II) at sufficiently low temperatures where rotons are not exited. In accordance with the recent experimental data, the phonon velocity dispersion is supposed to be positive, though small. Two different relaxation processes exist in the phonon system in this case: (i) the fast longitudinal relaxation establishing equilibrium phonon distribution along each direction in the momentum space with the temperature and the drift velocity depending on the direction; (ii) the slow transverse relaxation setting up equilibrium between different directions.Using the energy and momentum conservation and general principles of the irreversible thermodynamics we have derived the expression for the transverse relaxation operator. It appears to be a differential operator of the fourth order and depends on a function of the “phonon temperature” Θ that cannot be determined from the general consideration. We have calculated this function for the case of three-phonon collisions.Physical properties of the transverse relaxation operator are discussed and the corresponding boundary conditions are formulated. Several typical physical problems, both linear and nonlinear, which can be formulated in terms of the transverse relaxation operator, are enumerated. With the help of the diagram method the contribution of multiphonon collisions both in the longitudional and in the transverse relaxation is evaluated.     

The monadic theory and the “next world”

Suppose thatV is a model of ZFC andU ∈ V is a topological space or a richer structure for which it makes sense to speak about the monadic theory. LetB be the Boolean algebra of regular open subsets ofU. If the monadic theory ofU allows one to speak in some sense about a family ofκ everywhere dense and almost disjoint sets, then the second-orderV ^B-theory of ϰ is interpretable in the monadicV-theory ofU; this is our Interpretation Theorem. Applying the Interpretation Theorem we strengthen some previous results on complexity of the monadic theories of the real line and some other topological spaces and linear orders. Here are our results about the real line. Letr be a Cohen real overV. The second-orderV[r]-theory of ℵ₀ is interpretable in the monadicV-theory of the real line. If CH holds inV then the second-orderV[r]-theory of the real line is interpretable in the monadicV-theory of the real line. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author thanks the United States-Israel Binational Science Foundation for supporting the research.